11-1 Study Guide and Intervention

Study Guide and InterventionAreas of Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

11-111-1

© Glencoe/McGraw-Hill 611 Glencoe Geometry

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Areas of Parallelograms A parallelogram is a quadrilateral with both pairs ofopposite sides parallel. Any side of a parallelogram can be called a base. Each base has acorresponding altitude, and the length of the altitude is the height of the parallelogram.The area of a parallelogram is the product of the base and the height.

If a parallelogram has an area of A square units, Area of a Parallelogram a base of b units, and a height of h units,

then A � bh.

The area of parallelogram

ABCD is CD � AT.

Find the area of parallelogram EFGH.A � bh Area of a parallelogram

� 30(18) b � 30, h � 18

� 540 Multiply.

The area is 540 square meters.

Find the area of each parallelogram.

1. 2. 3.

Find the area of each shaded region.

4. WXYZ and ABCD are 5. All angles are right 6. EFGH and NOPQ arerectangles. angles. rectangles; JKLM is a

square.

7. The area of a parallelogram is 3.36 square feet. The base is 2.8 feet. If the measures ofthe base and height are each doubled, find the area of the resulting parallelogram.

8. A rectangle is 4 meters longer than it is wide. The area of the rectangle is 252 squaremeters. Find the length.

30 in.

18 in.

E F

GH

J K N O

PQLM

9 in.9 in.

9 in. 12 in.12 ft

3 ft

3 ft

3 ft8 ft 3 ft

2 ft6 ft

32 cm

16 cm

12 cm5 cm

W X

YZ

A B

D C

1.6 cm

1.6 cm60�

24 in.

16 ft

18 ft

18 m

30 mH G

FE

h

D T C

BA

b

ExampleExample

ExercisesExercises


Parallelograms on the Coordinate Plane To find the area of a quadrilateral onthe coordinate plane, use the Slope Formula, the Distance Formula, and properties ofparallelograms, rectangles, squares, and rhombi.

The vertices of a quadrilateral are A(�2, 2),B(4, 2), C(5, �1), and D(�1, �1).

a. Determine whether the quadrilateral is a square,a rectangle, or a parallelogram.Graph the quadrilateral. Then determine the slope of each side.

slope of A�B� � �4

2�

�

(�22)

� or 0

slope of C�D� � ��

�

11�

�

(�51)

� or 0

slope A�D� � ��

22�

�

(�(�

11))

� or �3

slope B�C� � ��51��

42

� or �3

Opposite sides have the same slope. The slopes of consecutive sides are not negativereciprocals of each other, so consecutive sides are not perpendicular. ABCD is aparallelogram; it is not a rectangle or a square.

b. Find the area of ABCD.From the graph, the height of the parallelogram is 3 units and AB � |4 � (�2)| � 6.A � bh Area of a parallelogram

� 6(3) b � 6, h � 3

� 18 units2 Multiply.

Given the coordinates of the vertices of a quadrilateral, determine whether thequadrilateral is a square, a rectangle, or a parallelogram. Then find the area.

1. A(�1, 2), B(3, 2), C(3, �2), and D(�1, �2)

2. R(�1, 2), S(5, 0), T(4, �3), and U(�2, �1)

3. C(�2, 3), D(3, 3), E(5, 0), and F(0, 0)

4. A(�2, �2), B(0, 2), C(4, 0), and D(2, �4)

5. M(2, 3), N(4, �1), P(�2, �1), and R(�4, 3)

6. D(2, 1), E(2, �4), F(�1, �4), and G(�1, 1)

x

y

OD C

BA

Study Guide and Intervention (continued)

Areas of Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

11-111-1

ExampleExample

ExercisesExercises

Skills PracticeArea of Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

11-111-1


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Find the perimeter and area of each parallelogram. Round to the nearest tenth ifnecessary.

1. 2.

3. 4.

5. 6.

Find the area of each figure.

7. 8.

COORDINATE GEOMETRY Given the coordinates of the vertices of a quadrilateral,determine whether it is a square, a rectangle, or a parallelogram. Then find thearea of the quadrilateral.

9. A(�4, 2), B(�1, 2), C(�1, �1), 10. P(�3, 3), Q(1, 3), R(1, �3),D(�4, �1) S(�3, �3)

11. D(�5, 1), E(7, 1), F(4, �4), 12. R(2, 3), S(4, 10), T(12, 10),G(�8, �4) U(10, 3)

8

222

2 2

2

263

3

4

6

6

1

1

1

1122

22

1

18.5 km

9 km

3.4 m

26 in.

22 in.

45�

14 yd

7 yd45�

5.5 ft

4 ft

60�

20 cm

30 cm60�

Study Guide and InterventionAreas of Triangles, Trapezoids, and Rhombi

NAME ______________________________________________ DATE ____________ PERIOD _____

11-211-2


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Areas of Triangles The area of a triangle is half the area of a rectangle with the samebase and height as the triangle.

If a triangle has an area of A square units, a base of b units,

and a corresponding height of h units, then A � �12

�bh.

Find the area of the triangle.

A � �12�bh Area of a triangle

� �12�(24)(28) b � 24, h � 28

� 336 Multiply.


Find the area of each figure.

1. 2.

3. 4.

5. 6.

7. The area of a triangle is 72 square inches. If the height is 8 inches, find the length of the base.

8. A right triangle has a perimeter of 36 meters, a hypotenuse of 15 meters, and a leg of 9 meters. Find the area of the triangle.

24 2418

10

21

1560

54

24

60�10

20

20

26

56

21

20 16

33

28 m

24 m

h

Z Y

X

b

ExampleExample

ExercisesExercises


Areas of Trapezoids and Rhombi The area of a trapezoid is the product of half theheight and the sum of the lengths of the bases. The area of a rhombus is half the product ofthe diagonals.

If a trapezoid has an area of A square units, bases of If a rhombus has an area of A square units andb1 and b2 units, and a height of h units, then diagonals of d1 and d2 units, then

A � �12

�h(b1 � b2). A � �12

�d1d2.

Find the area of the trapezoid.

A � �12�h(b1 � b2) Area of a trapezoid

� �12�(15)(18 � 40) h � 15, b1 � 18, b2 � 40

� 435 Simplify.


Find the area of each quadrilateral given the coordinates of the vertices.

1. 2.

3. 4.

5. 6.

7. The area of a trapezoid is 144 square inches. If the height is 12 inches, find the length ofthe median.

8. A rhombus has a perimeter of 80 meters and the length of one diagonal is 24 meters.Find the area of the rhombus.

28

12 24

13

1313

13

18

16

32

1632

12 60�

10

10

2020

40 m

18 m

15 m

d2 d1h

b2

b1


Areas of Triangles, Trapezoids, and Rhombi

NAME ______________________________________________ DATE ____________ PERIOD _____

11-211-2

ExampleExample

ExercisesExercises

Skills PracticeAreas of Triangles, Trapezoids, and Rhombi

NAME ______________________________________________ DATE ____________ PERIOD _____

11-211-2


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Find the area of each figure. Round to the nearest tenth if necessary.

1. 2. 3.

Find the area of each quadrilateral given the coordinates of the vertices.

4. trapezoid WXYZ 5. rhombus HIJKW(�5, 3), X(3, 3), Y(6, �3), Z(�8, �3) H(4, �3), I(2, �7), J(0, �3), K(2, 1)

Find the missing measure for each figure.

6. Trapezoid RSTU has an area of 7. Trapezoid JKLM has an area of 935 square centimeters. Find the 7.5 square inches. Find ML.height of RSTU.

8. Triangle ABC has an area of 9. Rhombus EFGH has an area of 1050 square meters. Find the 750 square feet. If EG is 50 feet,height of �ABC. find FH.

HE

F G

A C

B

60 m

M L

J K5 in.

2 in.

55 cmR U

TS 30 cm

34 cm28 cm

21 cm

40 cm

22 in.

12 in.25 in.

4 ft 3.5 ft6 ft

Study Guide and InterventionAreas of Regular Polygons and Circles

NAME ______________________________________________ DATE ____________ PERIOD _____

11-311-3


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Areas of Regular Polygons In a regular polygon, the segment drawn from the center of the polygon perpendicular to the opposite side is called the apothem. In the figure at the right, A�P� is the apothem and A�R� is the radius of the circumscribed circle.

If a regular polygon has an area of A square units,

Area of a Regular Polygon a perimeter of P units, and an apothem of a units,

then A � �12

�Pa.

U

V

R

A

PS

T

Verify the formula

A � �12�Pa for the regular pentagon above.

For �RAS, the area is

A � �12�bh � �

12�(RS)(AP). So the area of the

pentagon is A � 5��12��(RS)(AP). Substituting

P for 5RS and substituting a for AP, then

A � �12�Pa.

Find the area of regularpentagon RSTUV above if its perimeteris 60 centimeters.First find the apothem.The measure of central angle RAS is �36

50

� or 72. Therefore m�RAP � 36. The perimeteris 60, so RS � 12 and RP � 6.

tan �RAP � �RAP

P�

tan 36° � �A6P�

AP � �tan636°�

� 8.26

So A � �12�Pa � �

12�60(8.26) or 247.7.

The area is about 248 square centimeters.

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find the area of each regular polygon. Round to the nearest tenth.

1. 2. 3.

4. 5. 6.10.9 m

7.5 m10 in.

5��3 cm

15 in.10 in.14 m


Areas of Circles As the number of sides of a regular polygon increases, the polygon getscloser and closer to a circle and the area of the polygon gets closer to the area of a circle.

Area of a CircleIf a circle has an area of A square units and a radius of r units, then A � �r 2.

Circle Q is inscribed in square RSTU. Find the area of the shaded region.A side of the square is 40 meters, so the radius of the circle is 20 meters.

The shaded area is Area of RSTU � Area of circle Q� 402 � �r2

� 1600 � 400�

� 1600 � 1256.6� 343.4 m2

Find the area of each shaded region. Assume that all polygons are regular. Roundto the nearest tenth.

1. 2. 3.

4. 5. 6.

12 m

12 m

4 in.

16 m

4 m

2 in. 2 in.

Q

R S

TU40 m

40 m

O r


Areas of Regular Polygons and Circles

NAME ______________________________________________ DATE ____________ PERIOD _____

11-311-3

ExampleExample

ExercisesExercises

Skills PracticeAreas of Regular Polygons and Circles

NAME ______________________________________________ DATE ____________ PERIOD _____

11-311-3


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1. a pentagon with a perimeter of 45 feet

2. a hexagon with a side length of 4 inches

3. a nonagon with a side length of 8 meters

4. a triangle with a perimeter of 54 centimeters

Find the area of each circle. Round to the nearest tenth.

5. a circle with a radius of 6 yards

6. a circle with a diameter of 18 millimeters


7. 8.

9. 10.

5 cm4 ft

8 m

4 m

3 in.



1. a nonagon with a perimeter of 117 millimeters

2. an octagon with a perimeter of 96 yards

Find the area of each circle. Round to the nearest tenth.

3. a circle with a diameter of 26 feet

4. a circle with a circumference of 88 kilometers


5. 6.

7. 8.

DISPLAYS For Exercises 9 and 10, use the following information.A display case in a jewelry store has a base in the shape of a regular octagon. The length ofeach side of the base is 10 inches. The owners of the store plan to cover the base in blackvelvet.

9. Find the area of the base of the display case.

10. Find the number of square yards of fabric needed to cover the base.

9 m

25 ft

4.4 in.12 cm

Practice Areas of Regular Polygons and Circles

NAME ______________________________________________ DATE ____________ PERIOD _____

11-311-3


Identify Three-Dimensional Figures A polyhedron is a solid with all flatsurfaces. Each surface of a polyhedron is called a face, and each line segment where facesintersect is called an edge. Two special kinds of polyhedra are prisms, for which two facesare congruent, parallel bases, and pyramids, for which one face is a base and all the otherfaces meet at a point called the vertex. Prisms and pyramids are named for the shape oftheir bases, and a regular polyhedron has a regular polygon as its base.

Other solids are a cylinder, which has congruent circular bases in parallel planes, a cone,which has one circular base and a vertex, and a sphere.

Identify each solid. Name the bases, faces, edges, and vertices.

pentagonalprism

squarepyramid

pentagonalpyramid

rectangularprism

sphereconecylinder


Three-Dimensional Figures

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1

ExampleExamplea.

The figure is a rectangular pyramid. The base isrectangle ABCD, and the four faces �ABE, �BCE,�CDE, and �ADE meet at vertex E. The edges areA�B�, B�C�, C�D�, A�D�, A�E�, B�E�, C�E�, and D�E�. The verticesare A, B, C, D, and E.

E

A B

CD

b.

This solid is a cylinder. Thetwo bases are �O and �P.

P

O

ExercisesExercises


1. 2.

3. 4.

A B

CF

D E

P WQ

VZ X

U

SR

Y

T

S

R

Skills PracticeThree-Dimensional Figures

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1


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Draw the back view and corner view of a figure given each orthogonal drawing.

1. 2.


3.

4.

5.

S

R

F

AB C

DE

U T

WV

R S

XY

back view corner view

top view left view front view right view

back view corner view

top view left view front view right view

Study Guide and InterventionSurface Areas of Prisms

NAME ______________________________________________ DATE ____________ PERIOD _____

12-312-3


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Lateral Areas of Prisms Here are some characteristics of prisms.

• The bases are parallel and congruent.• The lateral faces are the faces that are not bases.• The lateral faces intersect at lateral edges, which are parallel.• The altitude of a prism is a segment that is perpendicular

to the bases with an endpoint in each base.• For a right prism, the lateral edges are perpendicular to the

bases. Otherwise, the prism is oblique.

Lateral Area If a prism has a lateral area of L square units, a height of h units, of a Prism and each base has a perimeter of P units, then L � Ph.

Find the lateral area of the regular pentagonal prism above if eachbase has a perimeter of 75 centimeters and the altitude is 10 centimeters.L � Ph Lateral area of a prism

� 75(10) P � 75, h � 10

� 750 Multiply.

The lateral area is 750 square centimeters.

Find the lateral area of each prism.

1. 2.

3. 4.

5. 6.

4 m16 m

4 in.

4 in.

12 in.

20 cm

10 cm 10 cm

12 cm9 cm

6 in.18 in.

15 in.

10 in.

8 in.4 m

3 m10 m

pentagonal prism

altitude

lateraledge lateral

face

ExampleExample

ExercisesExercises


Surface Areas of Prisms The surface area of a prism is thelateral area of the prism plus the areas of the bases.

Surface AreaIf the total surface area of a prism is T square units, its height is

of a Prismh units, and each base has an area of B square units and a perimeter of P units, then T � L � 2B.

Find the surface area of the triangular prism above.Find the lateral area of the prism.

L � Ph Lateral area of a prism

� (18)(10) P � 18, h � 10

� 180 cm2 Multiply.

Find the area of each base. Use the Pythagorean Theorem to find the height of thetriangular base.

h2 � 32 � 62 Pythagorean Theorem

h2 � 27 Simplify.

h � 3�3� Take the square root of each side.

B � �12� � base � height Area of a triangle

� �12�(6)(3�3�) or 15.6 cm2

The total area is the lateral area plus the area of the two bases.

T � 180 � 2(15.6) Substitution

� 211.2 cm2 Simplify.

Find the surface area of each prism. Round to the nearest tenth if necessary.

1. 2. 3.

4. 5. 6.

8 m

8 m8 m

8 in.

8 in.

8 in.

15 in.6 in.

12 in.

12 m

5 m6 m

3 m6 m

4 m

24 in.

10 in.

8 in.

6 cm

6 cm

6 cm

10 cmh


Surface Areas of Prisms

NAME ______________________________________________ DATE ____________ PERIOD _____

12-312-3

ExercisesExercises

ExampleExample

Skills PracticeSurface Areas of Prisms

NAME ______________________________________________ DATE ____________ PERIOD _____

12-312-3


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Find the lateral area of each prism.

1. 2.

3. 4.

Find the surface area of each prism. Round to the nearest tenth if necessary.

5. 6.

7. 8.

7

4

39

610

15

7

13

18

8

6

9

9

9

12

105

6

8

8

6

12

12

12

10

Study Guide and InterventionSurface Areas of Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

12-412-4


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Lateral Areas of Cylinders A cylinder is a solid whose bases are congruent circles that lie in parallel planes. The axisof a cylinder is the segment whose endpoints are the centers of these circles. For a right cylinder, the axis and the altitude of the cylinder are equal. The lateral area of a right cylinder is thecircumference of the cylinder multiplied by the height.

Lateral Area If a cylinder has a lateral area of L square units, a height of h units, of a Cylinder and the bases have radii of r units, then L � 2�rh.

Find the lateral area of the cylinder above if the radius of the baseis 6 centimeters and the height is 14 centimeters.L � 2�rh Lateral area of a cylinder

� 2�(6)(14) Substitution

� 527.8 Simplify.

The lateral area is about 527.8 square centimeters.

Find the lateral area of each cylinder. Round to the nearest tenth.

1. 2.

3. 4.

5. 6. 2 m

1 m12 m

4 m

20 cm

8 cm

6 cm

3 cm

3 cm

6 in.10 in.

12 cm

4 cm

radius of baseaxis

base

baseheight

ExercisesExercises

ExampleExample


Surface Areas of Cylinders The surface area of a cylinder is the lateral area of the cylinder plus the areas of the bases.

Surface Area If a cylinder has a surface area of T square units, a height of of a Cylinder h units, and the bases have radii of r units, then T � 2�rh � 2�r 2.

Find the surface area of the cylinder.Find the lateral area of the cylinder. If the diameter is 12 centimeters, then the radius is 6 centimeters.

L � Ph Lateral area of a cylinder

� (2�r)h P � 2�r

� 2�(6)(14) r � 6, h � 14

� 527.8 Simplify.

Find the area of each base.

B � �r2 Area of a circle

� �(6)2 r � 6

� 113.1 Simplify.

The total area is the lateral area plus the area of the two bases.T � 527.8 � 113.1 � 113.1 or 754 square centimeters.

Find the surface area of each cylinder. Round to the nearest tenth.

1. 2.

3. 4.

5. 6.

20 in.

8 in.

2 m 15 m

8 in.

12 in.

2 yd

3 yd

2 m

2 m

12 in.

10 in.

14 cm

12 cm

base

baselateral area


Surface Areas of Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

12-412-4

ExercisesExercises

ExampleExample

Skills PracticeSurface Areas of Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

12-412-4


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Find the surface area of a cylinder with the given dimensions. Round to thenearest tenth.

1. r � 10 in., h � 12 in. 2. r � 8 cm, h � 15 cm

3. r � 5 ft, h � 20 ft 4. d � 20 yd, h � 5 yd

5. d � 8 m, h � 7 m 6. d � 24 mm, h � 20 mm

Find the surface area of each cylinder. Round to the nearest tenth.

7. 8.

Find the radius of the base of each cylinder.

9. The surface area is 603.2 square meters, and the height is 10 meters.

10. The surface area is 100.5 square inches, and the height is 6 inches.

11. The surface area is 226.2 square centimeters, and the height is 5 centimeters.

12. The surface area is 1520.5 square yards, and the height is 14.2 yards.

4 m

8.5 m

5 ft

7 ft

Study Guide and InterventionSurface Areas of Pyramids

NAME ______________________________________________ DATE ____________ PERIOD _____

12-512-5


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Lateral Areas of Regular Pyramids Here are some properties of pyramids.• The base is a polygon.• All of the faces, except the base, intersect in a common point known as the vertex.• The faces that intersect at the vertex, which are called lateral faces, are triangles.

For a regular pyramid, the base is a regular polygon and the slant height is the height of each lateral face.

Lateral Area of a If a regular pyramid has a lateral area of L square units, a slant height of � units, Regular Pyramid and its base has a perimeter of P units, then L � �1

2�P�.

The roof of a barn is a regular octagonal pyramid. The base of the pyramid has sides of 12 feet,and the slant height of the roof is 15 feet. Find the lateral area of the roof.The perimeter of the base is 8(12) or 96 feet.

L � �12�P� Lateral area of a pyramid

� �12�(96)(15) P � 96, � � 15

� 720 Multiply.

The lateral area is 720 square feet.

Find the lateral area of each regular pyramid. Round to the nearest tenth ifnecessary.

1. 2.

3. 4.

5. 6.

45� 12 yd60�18 in.

60� 6 ft

42 m

20 m

3.5 ft

10 ft

8 cm

8 cm

8 cm

15 cm

base

lateral edgelateral face

vertex

slant height

ExercisesExercises

ExampleExample


Surface Areas of Regular Pyramids The surface area of a regular pyramid is the lateral area plus the area of the base.

Surface Area of aIf a regular pyramid has a surface area of T square units,

Regular Pyramida slant height of � units, and its base has a perimeter of P units and an area of B square units, then T � �

12

�P� � B.

For the regular square pyramid above, find the surface area to thenearest tenth if each side of the base is 12 centimeters and the height of thepyramid is 8 centimeters.Look at the pyramid above. The slant height is the hypotenuse of a right triangle. One leg ofthat triangle is the height of the pyramid, and the other leg is half the length of a side of thebase. Use the Pythagorean Theorem to find the slant height �.

�2 � 62 � 82 Pythagorean Theorem

� 100 Simplify.

� � 10 Take the square root of each side.

T � �12�P� � B Surface area of a pyramid

� �12�(4)(12)(10) � 122 P � (4)(12), � � 10, B � 122

� 384 Simplify.

The surface area is 384 square centimeters.

Find the surface area of each regular pyramid. Round to the nearest tenth ifnecessary.

1. 2.

3. 4.

5. 6.12 yd

10 yd

12 cm 13 cm

6 in.8.7 in. 15 in.

60�

10 cm

45�

8 ft

15 cm

20 cm

lateral edge

baseslant height height


Surface Areas of Pyramids

NAME ______________________________________________ DATE ____________ PERIOD _____

12-512-5

ExampleExample

ExercisesExercises

Skills PracticeSurface Area of Pyramids

NAME ______________________________________________ DATE ____________ PERIOD _____

12-512-5


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Find the surface area of each regular pyramid. Round to the nearest tenth ifnecessary.

1. 2.

3. 4.

5. 6.

7. 8.

16 in.

20 in.

18 m

12 m

6 yd

7 yd

9 mm

6 mm

14 ft

12 ft9 m

10 m

20 in.

8 in.4 cm

7 cm

Study Guide and InterventionSurface Areas of Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

12-612-6


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6Lateral Areas of Cones Cones have the following properties.• A cone has one circular base and one vertex.• The segment whose endpoints are the vertex and

the center of the base is the axis of the cone.• The segment that has one endpoint at the vertex, is

perpendicular to the base, and has its other endpoint on the base is the altitude of the cone.

• For a right cone the axis is also the altitude, and any segment from the circumference of the base to the vertex is the slant height �. If a cone is not a right cone, it is oblique.

Lateral Area If a cone has a lateral area of L square units, a slant height of � units, of a Cone and the radius of the base is r units, then L � �r�.

Find the lateral area of a cone with slant height of 10 centimeters and a base with a radius of 6 centimeters.L � �r� Lateral area of a cone

� �(6)(10) r � 6, � � 10

� 188.5 Simplify.

The lateral area is about 188.5 square centimeters.

Find lateral area of each circular cone. Round to the nearest tenth.

1. 2.

3. 4.

5. 6.

8 yd

16 yd12 in.

5 in.

26 mm

20 mm

6��3 m

60�

9 cm

15 cm

3 cm4 cm

6 cm10 cm

axis

base base� slant height

right coneoblique cone

altitudeV V

ExercisesExercises

ExampleExample


Surface Areas of Cones The surface area of a cone is the lateral area of the cone plus the area of the base.

Surface Area If a cone has a surface area of T square units, a slant height of of a Right Cone � units, and the radius of the base is r units, then T � �r� � �r 2.

For the cone above, find the surface area to the nearest tenth if theradius is 6 centimeters and the height is 8 centimeters.The slant height is the hypotenuse of a right triangle with legs of length 6 and 8. Use thePythagorean Theorem.

�2 � 62 � 82 Pythagorean Theorem

�2 � 100 Simplify.

� � 10 Take the square root of each side.

T � �r� � �r2 Surface area of a cone

� �(6)(10) � � � 62 r � 6, � � 10

� 301.6 Simplify.

The surface area is about 301.6 square centimeters.

Find the surface area of each cone. Round to the nearest tenth.

1. 2.

3. 4.

5. 6.

60�

8��3 yd26 m

40 m

4 in.

45�12 cm

13 cm

5 ft

30�9 cm

12 cm

�slant height

height

r


Surface Areas of Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

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ExercisesExercises

ExampleExample

Skills PracticeSurface Areas of Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

12-612-6


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6Find the surface area of each cone. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

7. Find the surface area of a cone if the height is 12 inches and the slant height is 15 inches.

8. Find the surface area of a cone if the height is 9 centimeters and the slant height is 12 centimeters.

9. Find the surface area of a cone if the height is 10 meters and the slant height is 14 meters.

10. Find the surface area of a cone if the height is 5 feet and the slant height is 7 feet.

4 yd6 yd

7 cm

22 cm

17 mm

9 mm

8 in.

21 in.

10 ft

25 ft14 m

5 m

Study Guide and InterventionSurface Areas of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

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Properties of Spheres A sphere is the locus of all points that are a given distance from a given point called its center.

Here are some terms associated with a sphere.• A radius is a segment whose endpoints are the

center of the sphere and a point on the sphere.• A chord is a segment whose endpoints are points

on the sphere.• A diameter is a chord that contains the sphere’s

center.• A tangent is a line that intersects the sphere in

exactly one point.• A great circle is the intersection of a sphere and

a plane that contains the center of the sphere.• A hemisphere is one-half of a sphere. Each great

circle of a sphere determines two hemispheres.

Determine the shapes you get when you intersect a plane with a sphere.

The intersection of plane M The intersection of plane N The intersection of plane Pand sphere O is point P. and sphere O is circle Q. and sphere O is circle O.

A plane can intersect a sphere in a point, in a circle, or in a great circle.

Describe each object as a model of a circle, a sphere, a hemisphere, or none of these.

1. a baseball 2. a pancake 3. the Earth

4. a kettle grill cover 5. a basketball rim 6. cola can

Determine whether each statement is true or false.

7. All lines intersecting a sphere are tangent to the sphere.

8. Every plane that intersects a sphere makes a great circle.

9. The eastern hemisphere of Earth is congruent to the western hemisphere.

10. The diameter of a sphere is congruent to the diameter of a great circle.

OPO

QN

O

PM

R�S� is a radius. A�B� is a chord.S�T� is a diameter. VX�� is a tangent.The circle that contains points S, M, T, and N isa great circle; it determines two hemispheres.

chord tangent

great circle

diameter

radius

S

T

W

RN

B

V

X

A

M

sphere

center

ExampleExample

ExercisesExercises


Surface Areas of Spheres You can think of the surface area of a sphere as the total area of all of the nonoverlapping strips it would take to cover thesphere. If r is the radius of the sphere, then the area of a great circle of thesphere is �r2. The total surface area of the sphere is four times the area of agreat circle.

Surface Area of a Sphere

If a sphere has a surface area of T square units and a radius of r units, then T � 4�r 2.

Find the surface area of a sphere to the nearest tenth if the radius of the sphere is 6 centimeters.T � 4�r2 Surface area of a sphere

� 4� � 62 r � 6

� 452.4 Simplify.

The surface area is 452.4 square centimeters.

Find the surface area of each sphere with the given radius or diameter to thenearest tenth.

1. r � 8 cm 2. r � 2�2� ft

3. r � � cm 4. d � 10 in.

5. d � 6� m 6. d � 16 yd

7. Find the surface area of a hemisphere with radius 12 centimeters.

8. Find the surface area of a hemisphere with diameter � centimeters.

9. Find the radius of a sphere if the surface area of a hemisphere is 192� square centimeters.

6 cm

r


Surface Areas of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

12-712-7

ExercisesExercises

ExampleExample

Skills PracticeSurface Areas of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

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In the figure, A is the center of the sphere, and plane Tintersects the sphere in circle E. Round to the nearesttenth if necessary.

1. If AE � 5 and DE � 12, find AD.

2. If AE � 7 and DE � 15, find AD.

3. If the radius of the sphere is 18 units and the radius of �E is 17 units, find AE.

4. If the radius of the sphere is 10 units and the radius of �E is 9 units, find AE.

5. If M is a point on �E and AD � 23, find AM.

Find the surface area of each sphere or hemisphere. Round to the nearest tenth.

6. 7.

8. a hemisphere with a radius of the great circle 8 yards

9. a hemisphere with a radius of the great circle 2.5 millimeters

10. a sphere with the area of a great circle 28.6 inches

32 m7 in.

A

ED

T


In the figure, C is the center of the sphere, and plane Bintersects the sphere in circle R. Round to the nearesttenth if necessary.

1. If CR � 4 and SR � 14, find CS.

2. If CR � 7 and SR � 24, find CS.

3. If the radius of the sphere is 28 units and the radius of �R is 26 units, find CR.

4. If J is a point on �R and CS � 7.3, find CJ.

Find the surface area of each sphere or hemisphere. Round to the nearest tenth.

5. 6.

7. a sphere with the area of a great circle 29.8 meters

8. a hemisphere with a radius of the great circle 8.4 inches

9. a hemisphere with the circumference of a great circle 18 millimeters

10. SPORTS A standard size 5 soccer ball for ages 13 and older has a circumference of27–28 inches. Suppose Breck is on a team that plays with a 28-inch soccer ball. Find thesurface area of the ball.

89 ft6.5 cm

C

RS

B

Practice Surface Areas of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

12-712-7

Study Guide and InterventionVolumes of Prisms and Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1


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Volumes of Prisms The measure of the amount of space that a three-dimensional figure encloses is the volume of the figure. Volume is measured in units such as cubic feet, cubic yards, or cubic meters. One cubic unit is the volume of a cube that measures one unit on each edge.

27 cubic feet � 1 cubic yard

Volume If a prism has a volume of V cubic units, a height of h units, of a Prism and each base has an area of B square units, then V � Bh.

cubic foot cubic yard

Find the volume of the prism.

V � Bh Formula for volume

� (7)(3)(4) B � (7)(3), h � 4

� 84 Multiply.

The volume of the prism is 84 cubiccentimeters.

7 cm3 cm

4 cm

Find the volume of theprism if the area of each base is 6.3square feet.

V � Bh Formula for volume

� (6.3)(3.5) B � 6.3, h � 3.5

� 22.05 Multiply.

The volume is 22.05 cubic feet.

3.5 ft

base


ExercisesExercises

Find the volume of each prism. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

7 yd4 yd

3 yd

4 cm

6 cm

2 cm

1.5 cm

10 ft15 ft

12 ft

30�15 ft

12 ft

3 cm

4 cm

1.5 cm

8 ft

8 ft

8 ft


Volumes of Cylinders The volume of a cylinder is the product of the height and the area of the base. The base of a cylinder is a circle, so the area of the base is �r2.

Volume of If a cylinder has a volume of V cubic units, a height of h units, a Cylinder and the bases have radii of r units, then V � �r 2h.

r

h


Volumes of Prisms and Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1

Find the volume of the cylinder.

V � �r2h Volume of a cylinder

� �(3)2(4) r � 3, h � 4

� 113.1 Simplify.

The volume is about 113.1 cubiccentimeters.

4 cm

3 cm

Find the area of the oblique cylinder.

The radius of each base is 4 inches, so the area ofthe base is 16� in2. Use the Pythagorean Theoremto find the height of the cylinder.

h2 � 52 � 132 Pythagorean Theorem

h2 � 144 Simplify.

h � 12 Take the square root of each side.

V � �r2h Volume of a cylinder

� �(4)2(12) r � 4, h �12

� 603.2 in3 Simplify.

8 in.

13 in.

5 in.

h


ExercisesExercises

Find the volume of each cylinder. Round to the nearest tenth.

1. 2.

3. 4.

5. 6.

1 yd4 yd

10 cm

13 cm

20 ft

20 ft12 ft1.5 ft

18 cm2 cm2 ft

1 ft

Skills PracticeVolumes of Prisms and Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

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Find the volume of each prism or cylinder. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

Find the volume of each oblique prism or cylinder. Round to the nearest tenth ifnecessary.

7. 8.

5 in.

3 in.17 cm

18 cm

4 cm

6 yd

10 yd15 mm23 mm

16 in. 22 in.

34 in.

3 m

5 m

13 m

6 ft

8 ft

2 ft

18 cm

16 cm

8 cm

Study Guide and InterventionVolumes of Pyramids and Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2


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Volumes of Pyramids This figure shows a prism and a pyramid that have the same base and the same height. It is clear that the volume of the pyramid is less than the volume of the prism. More specifically,the volume of the pyramid is one-third of the volume of the prism.

Volume of If a pyramid has a volume of V cubic units, a height of h units, a Pyramid and a base with an area of B square units, then V � �1

3�Bh.

Find the volume of the square pyramid.

V � �13�Bh Volume of a pyramid

� �13�(8)(8)10 B � (8)(8), h � 10

� 213.3 Multiply.

The volume is about 213.3 cubic feet.

Find the volume of each pyramid. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6. 6 yd

8 yd

5 yd15 in.

15 in.

16 in.

18 ft

regularhexagon 6 ft

4 cm8 cm

12 cm

10 ft

6 ft15 ft

12 ft

8 ft

10 ft

8 ft

8 ft

10 ft

ExercisesExercises

ExampleExample


Volumes of Cones For a cone, the volume is one-third the product of the height and the base. The base of a cone is a circle, so the area of the base is �r2.

Volume of a Right If a cone has a volume of V cubic units, a height of h units, Circular Cone and the area of the base is B square units, then V � �1

3�Bh.

The same formula can be used to find the volume of oblique cones.

Find the volume of the cone.

V � �13��r2h Volume of a cone

� �13��(5)212 r � 5, h � 12

� 314.2 Simplify.

The volume of the cone is about 314.2 cubic centimeters.

Find the volume of each cone. Round to the nearest tenth.

1. 2.

3. 4.

5. 6.

16 cm

45�26 ft

20 ft

45�18 yd

20 yd30 in.

12 in.

8 ft

10 ft6 cm10 cm

12 cm

5 cm

r

h


Volumes of Pyramids and Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2

ExercisesExercises

ExampleExample

Skills PracticeVolumes of Pyramids and Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2


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2

Find the volume of each pyramid or cone. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

Find the volume of each oblique pyramid or cone. Round to the nearest tenth ifnecessary.

7. 8.

12 cm

6 cm

4 ft4 ft

6 ft

66�18 mm

25 yd

14 yd

25 m

12 m

8 in.10 in.

14 in.

4 cm7 cm

8 cm

5 ft5 ft

8 ft

Study Guide and InterventionVolumes of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

13-313-3


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Volumes of Spheres A sphere has one basic measurement, the length of its radius. If you know the radius of a sphere, you can calculate its volume.

Volume of a Sphere

If a sphere has a volume of V cubic units and a radius of r units, then V � �43

��r 3.

Find the volume of a sphere with radius 8 centimeters.

V � �43��r3 Volume of a sphere

� �43��(8)3 r � 8

� 2144.7 Simplify.

The volume is about 2144.7 cubic centimeters.

A sphere with radius 5 inches just fits inside a cylinder. What is the difference between the volume of thecylinder and the volume of the sphere? Round to the nearest cubic inch.The base of the cylinder is 25� in2 and the height is 10 in., so the volume of the cylinder is 250� in3. The volume of the sphere is �

43��(5)3

or �5030�� in3. The difference in the volumes is 250� � �

5030�� or about 262 in3.

Find the volume of each solid. Round to the nearest tenth.

1. 2. 3.

4. 5. 6.

7. A hemisphere with radius 16 centimeters just fits inside a rectangular prism. What isthe difference between the volume of the prism and the volume of the hemisphere?Round to the nearest cubic centimeter.

8 in. difference between volume of cube and volume of sphere

13 in.5 in.

8 cm

16 in.

6 in.

5 ft

5 in.

5 in.

5 in.5 in.

8 cm

r

ExercisesExercises

Example 1Example 1

Example 2Example 2


Solve Problems Involving Volumes of Spheres If you want to know if a spherecan be packed inside another container, or if you want to compare the capacity of a sphereand another shape, you can compare volumes.

Compare the volumes of the sphere and the cylinder. Determine which quantity is greater.

V � �43��r3 Volume of sphere V � �r2h Volume of cylinder

� �r2(1.5r) h � 1.5r

� 1.5�r3 Simplify.

Compare �43��r3 with 1.5�r3. Since �

43� is less than 1.5, it follows that

the volume of the sphere is less than the volume of the cylinder.

Compare the volume of a sphere with radius r to the volume of each figure below.Which figure has a greater volume?

1. 2.

3. 4.

5. 6.2a

r

3r

r

r3r

r

rr

rr2

2r

r1.5r


Volumes of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

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ExercisesExercises

ExampleExample

Skills PracticeVolumes of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

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Find the volume of each sphere or hemisphere. Round to the nearest tenth.

1. The radius of the sphere is 9 centimeters.

2. The diameter of the sphere is 10 inches.

3. The circumference of the sphere is 26 meters.

4. The radius of the hemisphere is 7 feet.

5. The diameter of the hemisphere is 12 kilometers.

6. The circumference of the hemisphere is 48 yards.

7. 8.

9. 10.

14.4 m

4.5 in.

94.8 ft16.2 cm

Documents

11-1 Study Guide and Intervention